Semicircle angles and tangent radii
Use the two Core circle theorems with their required conditions and geometric reasons.
Core and Extended.
Before you begin
- Identify a circle's diameter, radius and tangent.
- Use triangle angle sums.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which chord is a diameter?
What you will learn
- Identify the right angle subtended by a diameter.
- Use the perpendicular radius at a tangent's contact point.
- Combine these facts with triangle angle sums.
A diameter makes a right angle at the circle
If A and B are the endpoints of a diameter and C is another point on the circumference, angle ACB is 90°. This is the angle in a semicircle theorem. The angle's vertex is on the circumference and the opposite side AB must pass through the centre; a random chord does not guarantee a right angle.
If angle CAB is 32°, the remaining angle ABC is 180 − 90 − 32 = 58°. Write both reasons: angle ACB is a right angle because AB is a diameter, and the third angle follows from the triangle total. The triangle can be rotated or drawn with C below AB without changing these conditions.
The radius is perpendicular at contact
A tangent touches the circle at T. The radius OT from centre O to that precise contact point meets the tangent at 90°. A line from the centre to a different circumference point is not guaranteed perpendicular to the tangent. Add the correct radius to a diagram before using the theorem.
For an external point P on the tangent, triangle OTP has a right angle at T. If angle OPT is 38°, angle POT is 52° by the triangle sum. These are the two circle-angle rules required by Core. Extended adds further theorems in the next lesson; do not use a theorem name without identifying the chord, arc or contact point it refers to.
Pause and explain
A tangent touches at T. Which line is perpendicular to it?
Worked example
AB is a diameter and C lies on the circle. Angle CAB = 32°. Find ACB and ABC with reasons.
Show the worked solution
- ACB = 90° because the angle subtended by diameter AB at the circumference is a right angle.
- Angles in triangle ABC total 180°.
- ABC = 180 − 90 − 32 = 58°.
Answer ACB = 90°; ABC = 58°.
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
AB is a diameter, C is on the circle and angle ABC = 41°. Find ACB and BAC.
Give me a hint
Use the diameter condition before subtracting from 180°.
Compare my reasoning
- ACB is 90° by the angle in a semicircle theorem.
- BAC = 180 − 90 − 41.
- Thus BAC = 49°; the triangle total checks.
ACB = 90°; BAC = 49°.
Look for these in your work
- Identified the angle opposite the diameter.
- Gave a triangle-sum reason for the second result.
PT is tangent at T to a circle with centre O. Angle OPT = 38°. Find OTP and POT.
Give me a hint
The radius OT meets the tangent PT at contact.
Compare my reasoning
- OTP = 90° because a radius is perpendicular to the tangent at contact.
- POT = 180 − 90 − 38.
- The central angle in this triangle is therefore 52°.
OTP = 90°; POT = 52°.
Look for these in your work
- Used the correct contact radius.
- Distinguished the triangle's vertices.
A student claims that angle ACB is 90° because A, B and C are on a circle. What extra condition is needed, and why?
Give me a hint
Not every inscribed triangle is right-angled.
Compare my reasoning
- The chord AB opposite angle C must be a diameter.
- That requires AB to pass through the circle's centre.
- Without this condition, the semicircle theorem cannot establish the right angle.
AB must be a diameter; three circumference points alone are insufficient.
Look for these in your work
- Checked the theorem's condition.
- Explained why appearance or circle membership alone is insufficient.
Common mistakes
- Treating every chord as a diameter.
- Placing the right angle at the centre for the semicircle theorem.
- Using a radius to the wrong contact point.
What can you explain now?
Is a tangent perpendicular to the radius at its contact point?
Compare with the explanation
Yes, the angle is 90°.
The radius must end at the tangent's exact point of contact; that is the condition of the theorem.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, sketch both Core circle theorems. Mark the conditions before writing 90° or finding any remaining angle.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources